+537 If a and b are real numbers, \(a^2b^3=\frac{32}{27}\), and \(\frac{a}{b^3}=\frac{27}{4}\), what is a+b?
+128475 a^2 b^3 = 32/27 (1)
a/b^3 = 27/ 4 ⇒ b^3 = (4/27) a (2)
Sub (2) into (1) and we have that
a^2 (4/27) a = 32/27 simplify
a^3 (4/27) = 32/27
a^3 = ( 32/27) (27/4)
a^3 = 32/4
a^3 = 8
a = 2 and
b^3 = (4/27) (2) = 8 /27
So....taking the cube root of both sides
b = 2/3
So
a + b = 2 + 2/3 = 8 / 3
